What is the greatest 3-digit base 8 positive integer that is divisible by 5?  (Express your answer in base 8.)
The greatest 3-digit base 8 positive integer is $777_8$, which is equal to $7 \cdot 8^2 + 7 \cdot 8 + 7 = 511$.  This number leaves a remainder of 1 when divided by 5, so we subtract 1, to get $\boxed{776_8}$.